Table Of ContentSynchronization in Gradient Networks
Xingang Wang,1,2 Ying-Cheng Lai,3 and Choy Heng Lai4,2
1Temasek Laboratories, National University of Singapore, 117508 Singapore
2Beijing-HongKong-SingaporeJoint Centre for Nonlinear & Complex Systems (Singapore),
National University of Singapore, Kent Ridge, 119260 Singapore
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3Department of Electrical Engineering,
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2 Department of Physics and Astronomy,
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Arizona State University, Tempe, Arizona 85287, USA
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4Department of Physics, National University of Singapore, 117542 Singapore
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(Dated: February8,2008)
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D Abstract
C
. Thecontradiction between the fact that many empirical networks possess power-law degree distribution
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l and the finding that network of heterogeneous degree distribution is difficult to synchronize has formed
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a paradox in the study of network synchronization. Surprisingly, we find that this paradox can be well
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v solved whenproper gradients areintroduced tothenetwork links, i.e. heterogeneous degree distribution is
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3 in favor of synchronization in gradient networks. We analyze the general properties of gradient networks
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and explore their functions in enhancing network sychronizability. Based on these understandings, we
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suppose the basic principles for constructing efficient gradient networks and propose a specific coupling
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n scheme as verification. Comparing to the previous asymmetric coupling schemes, the new scheme not
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n only possesses amuchstronger synchronizability but alsouses fewnetwork information. Moreover, under
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i the framework of gradient network, the factors which had been employed in former studies in improving
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r network synchronizability can be well unified and identified. The validity of our findings is verified by
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analytical estimates onthebehavior ofeigenvalues aswellasdirected simulations oncoupled nonidentical
oscillators. Our study therefore suggests that, in addition to the topology advantage, scale-free networks
alsomanifesttheirdynamical advantage givenpropergradients areconsidered.
PACSnumbers:89.75.-k,89.20.Hh,05.10.-a
1
I. INTRODUCTION
The study of complex networks has attracted a great deal of interest since the discoveries of
the small-world [1] and scale-free [2] properties in many natural and man-made networks [3,
4, 5]. While in initial studies the nodes and links of a network are treated as identical, recent
studies have extended to the heterogeneous networks of scaled nodes [6] and weighted links [7]
where many new features and properties are discovered. Meanwhile, in many practical systems
the scalars, which usually reflect the different characteristics among the nodes, and the weights,
whichusuallycharacterizetheinformationtransportcapacitiesonthelinks,arecloselycorrelated.
Typical examples include the co-authorship networks [8], where the scalar can be regarded as
the number of papers published by one researcher and the weight represents the collaboration
timesbetweentworesearchers,andtheworld-wideairlinenetwork(WWAN)[9],wherethescalar
can be the airport capacity and the weight represents the amount of transportation between two
airports. In bothcases thelinkswhichconnectnodes oflargerscalaroften assumelarger weights.
However, the latter is different to the former in that in WWAN the transportation is directed and,
in most cases, the mass flows in two directions are not balanced, i.e., there exists gradient on the
links. Thegradientnetworks,whicharedefinedasthedirectedgraphsformedbylocalgradientsof
ascalarfielddistributedonthenodes,areubiquitousinnatureandplayanimportantroleinmany
biological and technical systems [10, 11]. For example, it has been shown that the congestion
tendencyoftrafficnetworkscan bedrasticallyreduced whengradient isconsidered [10, 11].
Itiswellknownthatthecollectivebehaviorofcomplexsystemsisstronglyinfluencedbytheir
underlingcouplingstructures. Onetypicalexampleischaossynchronizationincomplexnetworks
[5, 12, 13]. Compare to the regular networks, the synchronizability of both the small-world and
scale-free networks (SFN) are drastically enhanced due to the decreased average distance [14].
However, as shown by the recent studies [15, 16, 17], the heterogeneous distribution of both the
node degree and the link weight could suppress the synchronizability. The contradiction between
the fact that many empirical networks have the property of heterogeneity and the finding that
synchronizabilityare suppressed in heterogeneous networks has formed a paradox in the studyof
networksynchronization. Thisparadoxhasstimulatedthesearchingofoptimalnetworkconfigura-
tioninSFN.Specifically,foraSFNofgiventopologyandtotalcouplingcost,peoplewanttoknow
how to distribute the couplings could efficiently promote the synchronizability [18, 19, 20, 21].
In Ref. [18] the authors proposed to distribute the incoming coupling strengths according to the
2
localinformationofnodedegree(hereafterwemarkitasM-scheme). Itisfoundthatthesynchro-
nizability is solely determined by the average degree, independent of the degree distribution and
the system size. In particular, under the conditionof uniform coupling capacity distribution,SFN
achievesitsmaximumsynchronizabilitywhichissuperiortonetworkofhomogeneousdegreedis-
tribution. In Ref. [20] the incoming coupling strengths are proposed to be distributed according
tothebetweennesscentralityoflinks(hereafterwemarkitasC-scheme), itisfoundthatsynchro-
nizabilityreaches itsmaximumonlywhenthedistributionsofthesetwoquantitiesmatch. Inboth
cases, the couplings are directed and in general they are not balanced, i.e., one direction weights
overanotherdirection.
As more and more evidences point to the important roles of gradient couplings, an interesting
question is: which kind of gradient will be more efficient in improving the network synchroniz-
ability? The answer relies on two parallel investigations: how to set the gradient direction and
how to distribute the gradient weight. In setting the gradient direction, an intuitive method is to
let the gradient start from the larger degree node and point to the smaller one [18, 19, 20]. But
analysis of non-diagonalizable networks suggest that this setting can be arbitrary, given no loops
in the gradient network [21]. In distributing the gradient weight, the answer is even diverse and
confusing, the proposed methods range from the uniform weight distribution [19] (hereafter we
marked it as H-scheme) to the distributions based on local information of node degree [18, 21]
and on global information of link betweenness [20]. Therefore the global picture of the function
ofgradient/asymmetriccouplingis stillnotclear and furtherstudyis needed.
In this work, we will investigate the problem of network synchronization from the gradient
network point of view and give a generic understanding to the functions of gradient/asymmetric
couplingsin improvingnetwork synchronizability. Undertheframework ofgradient network,not
only the previous results about asymmetric coupling can be well unified, but also the picture of
how to distribute the gradient direction and weight becomes clear and simple. For example, now
it is straightforward to figure out the necessary conditions for heterogeneous network to have a
stronger synchronizability than homogeneous network and, more importantly, to what extend the
gradientcould benefitsthenetwork synchronizability.
The rest of the paper will be organized as follows. In Section II we will introduce the idea
of gradient network and some of its basic properties. Based on the previous experiences and
the new understanding of gradient network, In Section III we will suggest the basic criteria for
constructing optimal networks and propose a new coupling scheme as application. In Section IV
3
wewillanalysisthepropertiesofthegradientnetworkemergesinthenewcouplingschemeand,in
SectionV,showitsefficiencyinimprovingnetworksynchronizabilitybythemethodofeigenvalue
analysis. In Section VI we willdiscuss the multipleeffects of gradient and show howthe optimal
gradient changes its value according to the network parameters. In Section VII we present the
simulationresults on coupled nonidentical chaotic oscillators. Finally we give the discussion and
conclusioninSection VIII.
II. GRADIENTNETWORK
For any pair of connected nodes, there are actually two directed couplings. When the weights
in two directions are equal, we say the couplings are symmetric, otherwise, the couplings are
asymmetric. To highlight the role of asymmetric couplings, a natural way is to separate the mu-
tual couplings into two parts: the symmetric part and the asymmetric part. While the symmetric
partreflects thecommonstrengththattwonodesaffect each other, theasymmetricpart represents
the dominant role that one node put to another node. Extend this kind of separation to the whole
network we will find that the original asymmetrically coupled network can be separated into two
subgraphs: one undirected symmetric network and one directed gradient network. Both two sub-
graphshavethesametopologyastheoriginalnetwork,butthecouplingdirectionandweighthave
been changed, as schematically shown in Fig. 1. Specially, the gradient network consists of only
directed links. While synchronization of symmetric network has been well explored in previous
studies,theseparationofgradientnetworkfromtheasymmetricnetworkcouldsimplifytheprob-
lem and make the analysiseasier. Now our attention solely turns to the study of gradient network
and investigateitseffects on networksynchronization.
Pleasenotethatthesoformedgradientnetworkisdifferenttothoseconventionalones. Conven-
tionally gradient network is defined as the collection of directed links pointingto each node from
whichever of its near neighbors has the highest/lowest scalar [10]. For network of N nodes and
averagedegree< k >,accordingtotheconventionaldefinitiontherewillbeonlyN directedlinks
inthegradientnetwork. Whileforthemutuallycouplednetwork,everylinkmayhaveaasymmet-
ric part and willcontributes onegradient link. Thus the gradient network generated from coupled
system actually has totally < k > ×N/2 directed links. By noting that the gradients pointing to
one node may have different weights and the behavior of this node is mainly determined by the
largest one (a specific parameter will be introduced later to balance the weight distribution), it is
4
= +
Weighted asymmetric network Weighted symmetric network Weighted gradient network
FIG. 1: The schematic diagram shows how to separate an weighted asymmetric network into a weighted
symmetricnetworkandaweightedgradient network.
reasonable to reduce the gradient network from < k > ×N/2 links to N links by only keeping
the largest gradient pointingto each node. In other words, for gradient network of heterogeneous
weightdistribution,itsmainfeaturecanbequalitativelycapturedbythereducedgradientnetwork
consisting of only the important gradients. With this simplification, the gradient network gen-
erated from coupled system will consist with those conventional ones, therefore many results in
previous studiescan be directed used here [10, 11]. Hereafter we will only analyze the properties
andfunctionsofthiskindofreducedgradientnetwork,whilethecorrectnessofthissimplification
willbeverified laterbyeigenvalueanalysisand direct simulations.
Toconcretetheideafurther,wedescribeinfollowingtheconventionalwayofconstructinggra-
dientnetwork[10]. ConsidercomplexnetworkG = G(V,E)whichhasN nodesV = {1,....,N}
and L links E = {1,...,L}. The set of edges E is specified by a adjacency matrix A = {a },
i,j
a = 1 if i and j are connected, otherwise a = 0, while a = 0. For a given node i of de-
i,j i,j i,i
gree k , the set of its neighbors is denoted by V = {j ∈ V | a = 1}. Consider also a scalar
i i i,j
field h = {h ,...,h } defined on the set of nodes V, therefore each node has a scalar value h
1 N i
associated to it. For node i of degree k , the gradient is defined as the directed link pointing to i
i
from whichever of its k neighbors has the highest scalar. If the neighbors have the same scalar
i
or if the node has the same scalar as all its neighbors, the selection will be random. Gradient
network is just constructed as the collection of all these directed links. By this method, the gen-
erated gradient network consists of N nodes and N links. Except the node of the largest scalar
5
h = max{h ,...,h }, which receives gradient from a nodeof smallerscalar than itself, all other
l 1 N
nodes only receive gradient from nodes of higher scalar than themselves. Therefore, except the
2-node loop formed by the largest scalar node l and one of its neighbornode j, all the other links
formed a tree structure with nodes l and j at the root. It has been shown that, for homogeneous
networkofrandomscalardistribution,thedegreedistributionofgradientnetworkfollowsapower
lawscaling,P(k) ∼ k−ς,with ς ≈ −1 [10].
Comparingtothoseconventionalgradientnetworks,thegradientnetworkgeneratedfromcou-
pled system possesses some new features. Firstly, the node scalar usually are not of random dis-
tribution,it has acloserelationto thepropertiesofthenode. Secondly,thegradient areweighted.
Gradient between nodes of larger scalar difference usually assumes a different weight to that of
smaller scalar difference. Finally, for sparsely connected networks we need to consider the prob-
lem of network breaking, i.e. the degeneration problem [10]. All these new features extend the
conventional concept on gradient network and arise new problems for investigation. Our mission
is just to characterize this kind of new gradient network and explore its functions in steering the
collectivebehaviorofcouplednetworks.
III. CONSTRUCTINGTHECOUPLINGMATRIX
Gradient, also known as bias, has been employed in nonlinear studies for many years and
proven to be a powerful technique in many fields such as chaos control and chaos synchroniza-
tion. Forexample,forcoupledoscillatorsonlatticeithasbeenshownthattheincreaseofgradient
couldgreatlyenhancethesynchronizabilityofthesystem[22]. Theasymmetriccouplingschemes
proposed in Refs. [18, 19, 20] suggest that gradient can be used to complex networks as well.
Gradient has been also employed in turbulence control, where the introduction of gradient could
significantlyimprovethecontrolefficiency [23]. Whiledifferent studyproposes different scheme
in setting the gradient, these schemes are nonetheless not unified. In particular, for complex net-
works, people are still not clear of the proper way of gradient configuration, i.e. how to set the
gradientdirectionandweight. Now,fromtheviewpointofgradientnetwork,thesestudiescanbe
wellunified and theprinciplesforconstructingoptimalnetwork can bevividlyportrayed.
Byreviewingthepreviousstudies,wecangetsomeimportantcluesinconstructingthematrix.
(1)Based on therecent findingsthat ’hubs’are firstly synchronizedthan ’nonhubs’and thefirstly
synchronized’hubs’act asthe”core” inpatternformationofcomplexnetworks[24, 25], itseems
6
plausible to build mutual links between ’hubs’ in the first place. In other words, the gradient
betweenthehubs,iftheyaredirectlyconnected,shouldbesmall. By thissettingwewishtobuild
an efficient channel between thehubsand to formthesynchronouscorequickly. (2) Toguarantee
that the synchronous manifold of the ”core” can be efficiently propagated to the ’nonhubs’ while
keeping the stability of the ’core’ itself [19], it is reasonable to set the gradient start from ’hubs’
and point to ’nonhubs’. According to this requirement we can set the gradient direction. (3) In
practical systemsthegradientspointingtoonenodegenerallyhavedifferentweight, neighborsof
higherscalarmaygeneratelargergradientthanthoseoflowerscalar. Todistinguishthisdifference,
itis reasonableto setthegradient weightbeingproportionaltothescalar potential. Settinginthis
way,thebehaviorofeachnodewillmorelikelytofollowitslargerdegreeneighbors. Therefore,in
determiningthefinalsynchronousstateofthewholenetwork,thehighdegreenodeshavealarger
contribution than the smaller degree nodes. (4) For most practical systems, each node has only
limited information of the whole network, e.g. the degree information of itself or its neighbors.
Whichmeansthat,inconstructingthecouplingmatrix,theconvenientmethodshouldemployonly
thelocalnetwork information[18, 19].
Based on the above understandings, we propose a new coupling scheme for network synchro-
nization. In particular,weconsidernetworksofcoupledchaoticoscillatorsfollowingequations
N
x. = F(x )−ε G H[x ], i = 1,...N, (1)
i i i,j j
Xj=1
where F(x ) governs the local dynamics of uncoupled node i, H[x] is a linear coupling function
i
and ε is the coupling strength. G is a zero rowsum coupling matrix with off diagonal entries
i,j
read
A kβ
G = i,j j , i,j = 1,...N, (2)
i,j N L kβ
j=1 i,j j
P
withAtheadjacencymatrixdefined inSec. II, k denotesthedegreeofnodej,andβ isatunable
j
parameter which will be used to adjust the gradient weight distribution. The diagonal entries are
unitG = 1.
i,i
IV. THEPROPERTIESOFGRADIENTNETWORK
We first investigate the properties of the gradient network formed by Eq. 2. For a pair of
connectednodesiandj,denotek (k )thedegreeofnodei(j)andV (V )thesetofitsneighbors,
ι j i j
7
thenthegradientfrom ito j reads
kβ kβ 1
∆G = G −G = i − j = [kβ kβ −kβ kβ], (3)
j,i j,i i,j kβ kβ C i l′ j l
l∈Vj l l′∈Vi l′ i,j lX′∈Vi Xl∈Vj
P P
with C = kβkβ. With the language of gradient network, it is straightforward to
i,j l∈Vj l′∈Vi l l′
P P
definethenodescalaras follows
h = kβ kβ. (4)
i i l
Xl∈Vi
Thisscalarwillbeusedtodeterminethegradientdirection,i.e. fromhigherscalartolowerscalar;
while the gradient weight is determined by the scalar potential ∆h = h − h and C , which
i,j i j i,j
depend on the degree situation of the node and its the neighbors. For SFN generated via the BA
model [2], there is no degree correlation between the nodes [27], i.e. the chance to find a large
degree neighbor or a small degree neighboris the same for any givennode. Specifically, we have
forthiskindofnetworktherelation kβ ∼ k . SubstitutethisrelationintoEq. 4 wehave
l∈Vi l i
P
h ∼ k1+β. (5)
i i
It can be found that, for positivevalueof β, larger degree node generally possesses a larger value
of h and the gradient is started from the larger degree nodeand point to the smallerdegree node,
i
while the contrary situation occurs when β < 0. (Note that with a slim chance the gradient may
flowfromsmallerdegreenodetolargerdegreenode. Thissituationhappenswhenasmallerdegree
nodes has very large degree neighbors. However, this chance is very small and, statistically, the
relationEq. 5isstillvalid,speciallyfornetworkofdenseconnections.) Therefore,wecancontrol
thegradientdirectionbychangingthesignofβ,whilecontrolitsweight,whichisproportionalto
thescalardifferenceas showninEq. 3, by itsabsolutevalue|β|.
The constructed gradient network consists of N nodes and < k > ×N/2 directed links. To
simplifytheanalysis,weassumeβ → ∞anddiscussthereducedgradientnetworkwhichconsists
of only N links. As we discussed in Sec. II, this reduction will not affect the main results we get
for gradient network, but will make the analysis much easier. Now we argue that the formed
gradient network is of forest structure (for obvious reason, we ignore the 2-node loop formed by
the largest degree node and one of its neighbors) and its out-degree follows a power law degree
distribution. To prove the forest structure, assume that on the contrary, there is a closed path
Φ = {∆G ,∆G ,...,∆G },l ≥ 3madeuponlyofdirectededgesselectedfromthegradient
2,1 3,2 l,l−1
network. Let j be the node on this path for which h = min{h ,h ,...,h }. Node j has exactly
j 1 2 m
8
-1 (a) 1.0 (b)
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SFN
RN Numerical results
0.8 Theoretical predictions
-3
10
Pout10-5 Pbreak00..46 break0.4
P
0.2
-7 0.2
10
0.0
0.0 0 20 40
-9 <K>
10
1 10 100 1000 0 5 10 15 20 25
Kout
FIG. 2: (Color online). For network of N = 210 nodes and mean degree < k >= 6. (a) The out-degree
distributionofthegradientnetworksgeneratedonSFNandRNsubstrates. Bothdistributionsfollowpower-
lawscaling withς ≈ −2.8forSFNsubstrate andς ≈ −1forRNsubstrate. (b)Theprobability ofnetwork
breaking as a function of degree heterogeneity γ. Inset plots the breaking probability as a function of the
mean degree < k >. It isfound that network tends to be broken as γ increases or< k >decreases. Each
dataisanaverageresultover100realizations.
twoneighborsonΦ, nodesj ±1,butonlyonegradient,∆G or∆G , pointingtoj. Since
j,j+1 j,j−1
h < h , both of the neighbors j ±1 will have their gradient edges pointinginto j. Since there
j j±1
are two edges, (j −1,j) and (j,j +1), and only one gradient edge will be accepted by j, one of
the edges must not be a gradient edge, and thus the loop is not closed, in contradiction with the
assumptionthatthereis aloopwithonlygradientedges.
Wegoon toestimatetheout-degreedistribution. Fornodeiwhichhasdegreek , assumenode
i
l ∈ V isoneofitsneighbors,thentheprobabilityforl toreceivegradientfromiisdeterminedby
i
two elements: the probability of h > h and the probability that h has the largest scalar among
i l i
the neighbors of l, i.e. h = max{h , j ∈ V }. Due to the zero degree correlation, the chance
i j l
to find a connection between i and l is p = k /(N −1); according to Eq. 5, the probability for
1 i
h > h equalsthatofk > k (herewediscussonlythecaseofβ > 0,whiletheanalysisforβ < 0
i l i l
is the similar), which is p = ki P(k)dk; the probability that h is the largest scalar among the
2 kmin i
neighbors of l is p == 1 −R(k − 1)p′ = 1 − (k − 1) kmaxP(k)dk, where p′ represents the
3 l 3 l ki 3
R
probabilitythatonelinkoflconnectstosomenodeoflargerdegreetoi. Therefore,theprobability
9
forl to contributeanout-degreetoi is
ki kmax
p = p p p = k /(N −1)· P(k)dk· 1−(k −1) P(k)dk . (6)
l 1 2 3 i l
Z (cid:20) Z (cid:21)
kmin ki
1
According to the network growth algorithm [2], we have kmax ≈ m × Nγ−1 and kmin = m (m
is the number of links associated to the new node in the BA growth model). Noting that the
contribution of out-degree may come from any node of degree smaller than k , substituting the
i
relation P(k) = Ck−γ = (γ −1)mγ−1k−γ into Eq. 6, we get the probability for each link of i to
become an out-degreelink is ki p NP(l)dl. Finally, thetotoal numberof out-degreefor nodei
kmin l
R
is
ki ki ki kmax
k = k · p NP(l)dl ≈ k · P(k)dk · 1−(l−1) P(k)dk Cl−γdl
out i l i
Z Z (cid:26) Z (cid:20) Z (cid:21)(cid:27)
kmin m m ki
ki ki kmax
= C2k · k · k−γdk · 1−C(l−1) k−γdk l−γdl (7)
i i
Z (cid:26) Z (cid:20) Z (cid:21)(cid:27)
m m ki
Denoting kik−γdk = aand kmaxk−γdk = b, wehave
m ki
R R
ki
k = C2k · {k ·a·[1−C(l−1)b]}l−γdl
out i i
Z
m
m2−γ −k2−γ
= aC2k · (1+Cb)a−Cb i (8)
i
(cid:26) γ −2 (cid:27)
Fork . k , Ca ≈ 1and Cb ≈ 0, wehave
i max
k ≈ k (9)
out i
Thereforetheout-degreedistributionofthegradientnetworkisthesametothedegreedistribution
ofSFN, which is
p ∼ P(k) ∼ k−γ. (10)
out i
The exponent we get from numerical simulation matches this prediction very well, as shown in
Fig. 2(a) where the fitted exponent is ς ≈ −2.8. Following the similar deduction, we can also
proof that the out-degree distribution of random network also follows a power law distribution,
but with exponent ς = −1. This relation is verified by numerical simulation again and the result
isplottedinFig. 2(a)by anothercurve.
Using a similar reasoning as for the forest structure, we can prove that there is no continuous
path which connects two local maxima of the scalar field h. This means that for each tree of the
gradient network there is only one local maximum scalar, and it is the only node which forms
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